Simplifying Multiplication Lessons

(Note * is used as the multiplication symbol.) When solving problems using algebra, it is often important to be able to
multiply algebraic terms which have variables. Examine this problem:

5x * 11x2

Clearly the methods for multiplying numbers in arithmetic do not apply here.
This lesson explains a method for doing this multiplication.

Before actually walking through such multiplication problems, we will present
a few concepts which are important to understand before learning the multiplication
technique.

Commutative Property

The Commutative Property of Multiplication is a statement or observation
about multiplication which indicates that the product of a multiplication
problem is the same, regardless of the order the terms were multiplied in.
For example,

2 * 10 = 20

and

10 * 2 = 20

Note that the product of 2 and 10 is 20 and that the product of 10 and 2 is
20, the products are the same. It turns out that this is true for real numbers in general. That is, the
order in which two numbers are multiplied does not affect the result.

This property can be extended to the case where we multiply more than 2 terms.
For example,

4 * 6 * 5 = 120

and

6 * 5 * 4 = 120

Notation

The next important concept is understanding the various styles of writing
multiplication in algebra. So far we have been using the * symbol in this lesson
to denote multiplication. There are several other mothods of showing multiplication
which are also acceptable. Consider the examples below. Each line shows one
way of writing "x squared times two".

Various Multiplication Notation Examples

x2 · 2

2 · x2

x2(2)

(x2)(2)

x2 * 2

You may recall using a "x" as the multiplication symbol in arithmetic. This
is generally avoided in algebra because x is the most common variable used in
expressions and equations. It would be extremely confusing if x were
used for both a variable and a multiplication sign.

In this lesson, we will continue to use * to represent multiplication because
it is easily entered with the keyboard, and because this notation is consistent
with our calculators and many handheld calculators.

Implied Exponents

Finally, it is important to understand the implied exponent on some variables.
When we consider a term like x3, we know that x has an exponent of
three. But what about the term x?

When a variable does not have a superscript which indicates its exponent, it
has an implied exponent of 1. Thus, x has an exponent of 1 and we can say that

x1 = x

Multiplying Variables

The first problem we will work on is

x * x

When two of the same variables are multiplied (in this case both x) the
answer consists of the varable with an exponent that is the sum of the other two
exponents.

In this case, both terms in the multiplication are x without a
superscripted exponent, therefore each has an implied exponent of 1.
We are multiplying so we will add the exponents, 1 + 1, to get 2.
Therefore, the answer is x to the second power:

x2

The next problem is

x * x4

This time we are multiplying x and x to the fourth power. The first
multiplier, x, doesn't have a visible exponent. As before we know that x has
an implied exponent of 1. The second multiplier, x4, has a visible
exponent, 4.

Both multipliers are the same letter, so we multiply by
adding the exponents. Since 1 + 4 = 5, the answer is:

x5

Multiplying Variables

More Than One Variable

In a problem like the one below, the terms cannot be multiplied by simply
adding exponents because each multiplier is a different letter variable.

y5 * x2

To show the answer to this problem, you simply remove the multiply sign
and put the terms next to each other, and in alphabetical order:

x2y5

Look over the next problem.

x2 * x3y2

As you can see, in this problem the x2 is multiplying a term which
has a variable x, and a variable y. The result of this multiplication is
found using a combination of the two previously presented techniques.

We begin by adding the exponents of x2 and x3: 2 + 3 =
5, therefore the product will contain x5.

Now since a y is present in the second term but not the first, it is simply
copied into the result. Thus the answer is:

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